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EXCHANGE 


DEFINITIVE  ELEMENTS 


OF 


COMET  1898X,  (BROOK'S) 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 
UNIVERSITY  OF  PENNSYLVANIA 


BY 
JOHNATHAN   T.  RORER 


IN  PARTIAL  FULFILMENT  OF  THE  REQUIREMENTS  FOR  THE  DEGREE 
OF  DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 

1910 


DEFINITIVE  ELEMENTS 


OF 


COMET  1898  X,  (BKOOK'S) 


A  THESIS 

PRESENTED  TO  THE  FACULTY  OF  THE  GRADUATE  SCHOOL  OF  THE 
UNIVERSITY  OF  PENNSYLVANIA 


BY 
JOHNATHAN   T.  RORER 


IN  PARTIAL  FULFILMENT  OF  THE  KEQUIREMENTS  FOR  THE  DEGREE 
OF  DOCTOR  OF  PHILOSOPHY 


PHILADELPHIA 

1910 


PRESS  OF 
THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER.   PA. 


DEFINITIVE  ELEMENTS  OF  COMET  1898  X, 
(BKOOK'S). 

Comet  1898  X  was  discovered  October  20th,  1898,  by  Dr.  W. 
K.  Brooks  of  Geneva,  N.  Y.  Observers  generally  recorded  the 
brightness  of  its  central  condensation  from  10m  to  llm.  Mr. 
Perrine  early  called  attention  to  the  similarity  of  the  elements  of 
1881  IV  (Schaeberle)  to  those  of  this  comet  (A.  J.,  XIX,  145). 

The  provisional  elements  assumed'  in  this  computation  were 
those  published  by  Hussey  (A.  J.,  XIX,  120),  based  on  obser- 
vations of  Oct.  21st,  23d  and  25th.  One  hundred  and  eighty- 
four  observations  were  collected  from  all  available  sources  and 
compared  with  an  ephemeris  computed  from  the  above  elements. 
To  perfect  these  and  lessen  the  residuals,  a  least  square  solution 
regarding  the  orbit  as  elliptic  was  then  made.  The  six  normal 
equations  thus  formed  lead  to  indeterminate  results.  It  was  at 
first  suspected  that  this  might  be  due  to  the  perturbations  of  the 
Earth  pad  Venus  ;  the  computation  of  these  perturbations  for 
the  time  covered  by  the  observations  showed  them  to  be  but  small 
fractions  of  a  second  of  arc  and  they  were  therefore  disregarded. 

It  was  next  determined  to  use  the  data  at  hand  to  obtain  the 
most  probable  parabolic  elements.  Three  new  normal  dates  were 
selected,  each  being  intermediate  between  successive  pairs  of 
former  normal  dates  and  the  normal  equations  then  used  to  obtain 
corrected  positions  for  these  dates. 

For  a  parabolic  orbit  from  these,  the  following  elements  result : 

r=  1898  Nov.  23.  16356  B.  M.  T. 

«  =  123°  31'  34".89    ,-,,...        ,  , ,       ^     . 

fl  =    96°20'41".26    Ecliptic  and  MeanEqumox 

»  =  140°  21'    5".05 
log.  g  =9.8786106 

The  usual  checks  were  applied  to  verify  the  work. 

3 

240959 


From  the  above  elements  the  positions  of  the  comet  were 
computed  for  ten  selected  dates  of  observation.  The  resulting 
residuals,  while  showing  a  marked  improvement  over  those  from 
Hussey's  elements,  were  deemed  capable  of  further  improvement, 
and  accordingly  the  residuals  for  the  date  of  each  observation 
were  found  by  interpolation  and  correction  of  the  previous  resi- 
duals. Ten  normal  places  were  then  formed  from  which  correc- 
tions to  the  above  elements  were  derived  by  a  second  least  square 
solution.  These  corrections  were  : 

ATT  =  -  242".  80 


Ai  =  +  51.36 
AI7  =  -  0.00206 
Ag  =  -  0.0000284, 

leading  to  the  elements  : 

T=  1898,  Nov.  23.  16150 
w  =  123°  29'  40".30 
ft  =  96°  18'  33".05 
i  =  140°  21'  56".41 
log  q  =  9.8785943 

After  the  completion  of  the  above  computation  forty-five  ad- 
ditional observations  became  available  ;  an  effort  was  therefore 
made  to  still  further  reduce  the  residuals  by  the  employment  of 
these  observations.  Using  the  above  elements  a  new  ephemeris 
was  computed  at  four-day  intervals  for  the  time  covered  by  the 
observations  and  from  this  by  interpolation  the  computed  place 
was  found  for  each  of  the  two  hundred  and  twenty-nine  observa- 
tions. The  residuals  thus  formed  were  plotted,  and  a  smooth 
curve  was  traced  representing  the  points  as  closely  as  possible. 
It  was  not  deemed  necessary  to  recompute  the  coefficients  of  the 
corrections  to  the  elements  used  in  the  normal  equations,  and  con- 
sequently the  same  normal  dates  were  used.  A  least  square 


solution  reduced  the  twenty  resulting  equations  to  five  normal 
equations  of  the  usual  form  : 

[1]  +38.083A7T—  68.580Ai2— 16.373A*— 21.725A2\+8.323Ag1  +2175.976=0 

[2]  —68.580       +124.173       +27.701     +38.568     —15.216  —3809.574=0 

[3]  —16.373      +  27.701        +47.936     +19.797     +  4.170  —1983.742=0 

[4]  —21.725      +  38.568       +19.797     +15.131       -  2.766  —1548.340=0 

[5]  +  8.323       -  15.216       +  4.170     —  2.766     +  3.548  +  255.329=0 

From  these  there  was  obtained  :  AH  =  —  31". 373 

106A^  =  Aft  =  +  94  .901 

ATT  =  +126.000 

(  Ai  =  -  136.805 

104AT=  A2;  =  +  559.540 

A  substitution  of  these  values  in  the  five  normal  equations  leads 
to  the  residuals : 

-0.14 
+  0.21 
-0.85 
-0.19 
-0.05 

Considering  the  size  of  the  constants,  the  above  residuals  are 
thought  to  be  satisfactory  and  the  following  elements  resulting 
from  the  above  corrections  are  adopted  as  the  final  values. 

FINAL  ELEMENTS. 
Nov.  23.  21745 

n  AO     or       it  fL    Ecliptic  and  Mean  Equinox 

o       ,     V*  of  1898.0 

i  =  140°  19'  39". 60 

log  q  =  9.8786488 

The  short  time  this  comet  was  observed,  36  days,  the  flatness 
of  the  arc  traversed,  the  slowness  of  the  motion,  and  the  difficul- 
ties arising  from  its  physical  appearance,  all  combine  to  produce 
indeterminate  results.  Much  effort  was  expended  to  obtain  a 


6 

greater  refinement  from  the  data,  and  the  present  publication  has 
been  long  delayed  in  the  hope  that  better  elements  might  be  ob- 
tained. Further  investigation,  however,  warrants  the  conclusion 
that  the  above  elements  may  be  considered  as  final. 

Since  the  above  calculations  were  completed  S.  Sharbe  has  pub- 
lished definitive  elements  of  this  comet  (A.  N.,  164,  p.  378).  Both 
parabolic  and  elliptic  elements  are  given,  the  latter  being  deemed 
definitive.  These  are  here  reproduced.  A  comparison  with  the 
elements  given  above  shows  substantial  agreement,  the  greatest 
differences  being  in  the  inclination  of  the  orbit  and  the  time  of 
perihelion  passage. 

Elements  of  the  Author  Sharbe's  Parabolic  Sharbe' s  Elliptic 

T  1898  Nov.  23.  21745  Nov.  23.  189594  Nov.  23.  195124 

w    123°  32'  17". 67  123°  31'  53". 96  123°  32'  23".70 

G   96°  18'  1".68  96°  18'  14".47  96°  18  12".46 

i    140°  19'  39//.60  140°  2(X  57".  50  140°  20'  51".  52 

log  q    9.8786488  9.8785281  9.8785038 

e    1.  1.  0.9997421 

The  value  of  de  given  by  Scharbe  is  —  0.0002579  ±  0.000- 
2600.  The  limits  of  error  are  evidently  so  large  that  there  is  no 
sufficient  reaaon  for  giving  preference  to  elliptic  elements.  The 
probable  errors  of  all  the  elements  indicate  the  indeterminate 
character  of  the  orbit,  to  which  reference  has  already  been  made. 


MAKERS 

SYRACUSE,  -  NY, 


